Related papers: Application of Fractional Derivative Operators to …
Fractional derivatives can be used to model time delays in a diffusion process. When the order of the fractional derivative is distributed over the unit interval, it is useful for modeling a mixture of delay sources. In some special cases…
Modeling of phenomena such as anomalous transport via fractional-order differential equations has been established as an effective alternative to partial differential equations, due to the inherent ability to describe large-scale behavior…
In this note, a numerical method based on finite differences to solve a class of nonlinear advection-diffusion fractional differential equation is proposed. The fractional operator considered here is the fractional Riemann-Liouville…
In this paper, a new fractional operator of variable order with the use of the monotonic increasing function is proposed in sense of Caputo type. The properties in term of the Laplace and Fourier transforms are analyzed and the results for…
This paper is concerned with the fractionalized diffusion equations governing the law of the fractional Brownian motion $B_H(t)$. We obtain solutions of these equations which are probability laws extending that of $B_H(t)$. Our analysis is…
When considering fractional diffusion equation as model equation in analyzing anomalous diffusion processes, some important parameters in the model related to orders of the fractional derivatives, are often unknown and difficult to be…
Fractional calculus, in allowing integrals and derivatives of any positive order (the term "fractional" kept only for historical reasons), can be considered a branch of mathematical physics which mainly deals with integro-differential…
A numerical method to solve the fractional diffusion equation, which could also be easily extended to many other fractional dynamics equations, is considered. These fractional equations have been proposed in order to describe anomalous…
Anomalous relaxation and diffusion processes have been widely characterized by fractional derivative models, where the definition of the fractional-order derivative remains a historical debate due to the singular memory kernel that…
This study unveils the time-space transforms underlying anomalous diffusion process. Based on this finding, we present the two hypotheses concerning the effect of fractal time-space fabric on physical behaviors and accordingly derive…
Fractional Dzherbashian-Nersesian operator is considered and three famous fractional order derivatives namely Riemann-Liouville, Caputo and Hilfer derivatives are shown to be special cases of the earlier one. The expression for Laplace…
Fractional diffusion and Fokker-Planck equations are widely used tools to describe anomalous diffusion in a large variety of complex systems. The equivalent formulations in terms of Caputo or Riemann-Liouville fractional derivatives can be…
This work concerns the asymptotic analysis of high-frequency wave propagation in randomly layered media with fast variations and long-range correlations. The analysis takes place in the 3D physical space and weak-coupling regime. The role…
This paper is devoted to describing a linear diffusion problem involving fractional-in-time derivatives and self-adjoint integro-differential space operators posed in bounded domains. One main concern of our paper is to deal with singular…
Fractional diffusion equations are widely used to describe anomalous diffusion processes where the characteristic displacement scales as a power of time. For processes lacking such scaling the corresponding description may be given by…
We analyze the well-posedness of an anisotropic, nonlocal diffusion equation. Establishing an equivalence between weighted and unweighted anisotropic nonlocal diffusion operators in the vein of unified nonlocal vector calculus, we apply our…
A physical-mathematical approach to anomalous diffusion may be based on fractional diffusion equations and related random walk models. The fundamental solutions of these equations can be interpreted as probability densities evolving in time…
The anomalous (i.e. non-Gaussian) dynamics of particles subject to a deterministic acceleration and a series of 'random kicks' is studied. Based on an extension of the concept of continuous time random walks to position-velocity space, a…
In physics, phenomena of diffusion and wave propagation have great relevance; these physical processes are governed in the simplest cases by partial differential equations of order 1 and 2 in time, respectively. By replacing the time…
In this paper we present stochastic foundations of fractional dynamics driven by fractional material derivative of distributed order-type. Before stating our main result we present the stochastic scenario which underlies the dynamics given…